Influence of the Wire Spatial Structure on the Distribution of Product and the Peak Overpressure of Shockwave Generated by the Electric Explosion

Abstract

The deposition energy and the peak overpressure of shockwaves are the leading engineering parameters of wire electric explosion technology applied to enhance oil recovery. The thicker Cu wire deposits more energy, which transforms into the shockwave efficiently. Therefore, the effects of three diameters (0.3, 0.4, and 0.5 mm) and hollow ratios (0, 0.5, and 0.7) on the electric explosion efficiency were studied by collecting pulse current, explosion products, and shockwaves during the test. All spatial structure designs of the wire depend on the skin effect parameters of the pulse discharge current. The results found that the peak overpressure of the shockwave soars with the increase of the hollow ratios when the diameter is constant. The range of the peak overpressure is 25.2~47.7 MPa. However, the correlation between shockwave and wire diameter changes from negative to positive with the increase of the hollow ratio from 0 to 0.7. The phase distribution deduced by the particle morphology and quantity distribution indicates that it is going to be uniform gradually with the hollow ratio rising from 0 to 0.7. When the extreme simplification is carried out without considering the magnetic diffusion process, it is indicated that the distribution of temperature and phase states along the wire radial is a Bessel function depending on the skin effect of the current density when three times the theoretical enthalpy drives the Cu wire. It means that the desired shockwave could be obtained efficiently by increasing the diameter and the hollow ratio of wire during a wire electric explosion.

1. Introduction

Wire electric explosion technology is a unique auxiliary technology to stimulate oil and gas well stimulation. Compared with conventional acidizing and hydraulic fracturing technology, electric explosion technology has the advantages of a simple construction process, short construction period, and environmental protection. Since the 1960s, there has been much research on this technology’s basic theory and principle prototype [1,2,3,4,5].

The study of electrical load resistivity is the theoretical basis of wire explosion. The results of high-purity Cu by experiments at high pressures show that the antagonistic effects of pressure and temperature interpret the resistivity of solid and liquid [6]. The resistivity of neutral copper gas is measured by the dark pause effect in a wire explosion [7]. Meanwhile, the weakening effects of water resistivity and phase state on an underwater electric explosion are also studied [8,9]. Shockwaves generated in electrical explosion tests have also been extensively studied. The peak overpressure of the shockwave is discussed by conducting a wire explosion under the different spatial arrangements and ambient pressure [10,11,12]. The magnitude and efficiency of the deposited energy are the fundamental reasons for experimental parameters’ effects on the shockwave’s peak overpressure in the process of a wire electric explosion [13,14,15,16]. However, quantitatively determining the process by which an electric wire explosion achieves a target shockwave is complex. The temporal and spatial evolution of a wire electric explosion is studied through optical tests [17], and the result parameters of the investigation are estimated by an empirical approach [18,19].

Meanwhile, the implementation scheme of electric explosion technology is related to the micromorphology of its nanoproducts [20]. In addition, the dispersive property of an electric explosion was used to prepare binary and complex nanoalloys [21,22]. The spatial–temporal distribution of the wire’s inner electromagnetic field generated by the pulsed current is a basis for pulsed power technology, such as a wire electric explosion. Studies on the skin effect show that the current density satisfies the Bessel function distribution on the cross-section of the wire when a pulsed electromagnetic field is loaded [23,24,25]. Furthermore, the magnetic field distribution calculated by combining the axial measurement and laser ablation technology indicates that the magnetic field is 5~15 T at a radial position of 5~7 mm [26]. The investigation of plasma evolution dynamics of the exploding thin Al liner shows that the effective current radius gets bigger with time because of the time-varying magnetic diffusion process [27].

In order to make the shockwaves generated by electric explosion technology reach the index of overpressure and deposition efficiency required by petroleum engineering, they need to match the driving energy and wire spatial structure to overcome or alleviate the influence of the skin effect. In this paper, the hollow cylindrical copper wire is proposed as an electric explosion medium when the circuit parameters are controllable and fixed. The correlation between the phase state distribution of products and the peak overpressure of the shockwave is studied to match the driving energy and wire spatial structure during the electric explosion process. Moreover, through the qualitative derivation of the spatial–temporal evolution of a wire electric explosion, the design of the wire spatial structure blunts the harmful effects and takes advantage of the beneficial effects to obtain the ideal peak overpressure of a shockwave.

2. Materials and Methods

2.1. Composition of the Experimental Rig

The underwater wire electric explosion test rig is shown in Figure 1. The main body of the test rig includes a pulse power generator (high-voltage DC (direct current) power supply, Marx generator, and switch), an electrical pulse-shockwave energy conversion device (explosion bracket, electrode, consumable Cu wire, cable, and test tank), a signal acquisition device (Rogowski coil, high voltage divider, shockwave probe, control module, and oscilloscope), and a collection unit (silica substrates).

The high-voltage DC power supply charges the Marx generator to store energy. The trigger switch transmits the pulse current to the energy conversion device. The pulse current drives the consumable Cu wire to make a rapid phase-change electrical explosion. Moreover, the electric explosion shockwave is diffused and propagated in the water. The Rogowski coil measures the pulse current signal of the electric explosion circuit. The high-voltage divider measures the voltage signal to earth at both ends of the wire. The shockwave probe measures the peak overpressure signal on the horizontal line across the geometric center of the wire. The current, voltage, and peak overpressure signals are imported into the high-frequency oscilloscope. The explosive products are collected by a silica substrate located in the direction of the shockwave propagation. The version and the performance parameters of a signal detection device and product collection device are shown in

Table 1. Performance parameters of signal detection device and product collection device.

Device	Version	Performance Parameters
Rogowski coil	Pearson-101	0~50 kA	Rise time 0.1 µs
High-voltage divider	SGB-50C	0~50 kV	Precision 0.5%
Shockwave probe	PCB-109C11	0~552 MPa	Rise time < 2 μs
Collection unit	Silica substrates	15 × 15 mm2	Polished Ra < 2 nm
Oscilloscope	DSO-2014A	0~100 MHz	Sample Rate 2 GSa/s

.

2.2. Model of Wire Electrical Explosion Circuit

In the pulse power system of wire explosion, the discharge circuit is the classical RLC circuit model [28]. Using Kirchhoff’s voltage law, the discharge circuit equation governing the differential equation is Equation (1).

$$L_h\frac{dI}{dt}+R_{h}I+\frac{1}{C}\int Idt=U$$

(1)

The capacitor, C, in the system is the primary energy storage component. The loop capacitance value can be simplified and is equivalent to the value of the energy storage capacitor. The discharge circuit resistance, R_h, and the circuit inductance, L_h, are affected mainly by the characteristics of the discharge circuit and its spatial arrangement. When $R_h\ll 2\sqrt{L_h/C}$, the impulse current, I(t), and its circuit parameters can be simplified as Equations (2) and (3). T represents the first approximate period of the impulse current, I_P1 and I_P2 represent the first and second peaks of the impulse current, respectively, and α represents the decay rate of the impulse current.

$$I\left(t\right)=U\sqrt{\frac{C}{L_h}}{e}^{-\alpha t}\sin {\omega }_{0}t$$

(2)

$$\left\{\begin{array}{c}T\approx 2\pi \sqrt{L_{h}C}\\ I_{P2}=e^{-\alpha T}{I}_{P1}\\ \alpha =R_h/2L_h\end{array}\right.$$

(3)

The test rig was directly short-circuited and discharged. Moreover, the discharge loop performance of the Marx generator was debugged and designed. The 5-stage Marx generator was used for the discharge test with a 3 kV charging voltage. The Marx generator has a single-stage capacity of 20 μF. For charging and discharging of the Marx generator, the equivalent capacities are 100 μF and 4 μF, respectively. According to the impulse current curve of 5 tests, the average value of the oscillation cycle and loop resistance, R_h, and inductance, L_h, through the calculation results of Equation (3) are as shown in

Table 2. Basic performance parameters of the circuit (5-stage Marx).

Test Voltage U/kV	First Current Peak IP1/kA	Second Current Peak IP2/kA	Approximate Period T/μs	Loop Resistance Rh/mΩ	Loop Inductance Lh/μH
3.00	20.80	12.21	15.9	107.2 ± 3.2	1.6 ± 0.1

.

2.3. Experimental Parameter Design

The spatial structure of the wire is a cylinder with a proportionally hollow interior. The hollow ratio, a, represents the ratio between the inner diameter, d, and outer diameter, D. The hollow structure of the wire and one of the physical objects are shown in Figure 2.

The length of the wire is a constant in this study because the current density is invariable in the length direction. A shorter wire requires a lower test voltage when the capacitance value of the system is constant. Considering the perforation hole size of the oil well affected by a shockwave in engineering practice, L represents the length of the wire, which is selected as 30.00 mm. The shockwave maintains almost the same cylinder shape, irrespective of the wire cross-section.

The electric explosion of metal wire is a physical phase transition process with a breakneck development speed. For the electric explosion of pure copper metal, the theoretical enthalpy value is the lowest energy required during the rapid transformation of pure copper wire from a condensed state to a plasma state.

Table 3. Thermochemical parameters of pure copper.

Material	Heat of Solid HS/kJ·mol−1	Latent Heat Fusion HF/kJ·mol−1	Heat of Liquid HL/kJ·mol−1	Latent Heat Vaporization HV/kJ·mol−1	Enthalpy Change Htotal/kJ·mol−1
Cu	29.66	13.14	48.78	300.68	392.26

shows the enthalpy value parameters of Cu(ref) in NIST-JANAF [29]. The theoretical total enthalpy value is 392.26 kJ·mol⁻¹.

Considering the influence of energy dissipation and the spatial instability during the underwater wire electric explosion process, the Marx generator’s pre-stored energy is set to three times the theoretical enthalpy value of pure copper wire. To obtain a high-power impulse current, the Marx generator is used to increase an impulse current and compress pulse discharge time. When the electric pulse is introduced into the electric explosive wire consumable, the current first forms on the surface due to the skin depth effect, and then the current density diffuses inward to the center with the evolution of the pulse. Therefore, slightly increasing the diameter of the wire and making it hollow can optimize the electrical explosion process. In this paper, the changes of energy deposition and shockwave pressure of the electric explosion of wire were studied with a different outer diameter, D, and wire hollow ratio, a, when three times the energy was injected. The driving energy, E_store, and voltage, U, required for different diameters and hollow ratios in the pure copper test are shown in

Table 4. Energy and voltage parameters for different diameters and hollow ratios.

Diameter D/mm	Hollow Ratio a (d/D)	Driving Energy Estore/J	Voltage U/kV
0.3	0	349.75	2.64
	0.5	262.31	2.29
	0.7	178.37	1.89
0.4	0	621.78	3.52
	0.5	466.33	3.05
	0.7	317.11	2.52
0.5	0	971.54	4.41
	0.5	728.65	3.82
	0.7	495.48	3.15

. The driving energy, E_store, is a physical quantity determined indirectly. In Equation (4), all parameters except the wire length, L, are fixed, and ρ and M represent the density and molar mass of copper, respectively. Because the wire length is measured with Vernier calipers, the energy calculation results retain five significant digits. The voltage, U, is a physical quantity estimated by driving energy and determined by measuring the instrument directly. The adjustment accuracy of the high-voltage DC power supply can only reach two decimal places.

$$E_{store}=\frac{3\pi }{4}\frac{\left(1-a^2\right)D^{2}L\rho }{M}{H}_{total}=\frac{1}{2}C{U}^2$$

(4)

In the electric explosion test, the conductivity of water has influence on the energy conversion process and the result of the final shockwave pressure test. In a published study [8], saline solutions with different conductivity are configured for electrical explosion tests. The experimental results show that, when the conductivity of the solution is less than the threshold of 5 mS/cm, the peak pressure of the shockwave will not be affected. At the location of the laboratory, the measured electrical conductivity of tap water is about 0.9 (±0.02) mS/cm when the temperature is 25 °C. Therefore, tap water is the choice as the research medium under the premise of not affecting the test results. The effect of repeated discharge experiments on water conductivity should also be eliminated. Using one of the experimental parameters in Table 4, the discharge process is repeated 10 times at intervals of 10 min. The deviation of water conductivity after the test is less than 0.04 mS/cm. Therefore, when tap water is used as the experimental medium, the experimental results are not affected.

2.4. Trainable Weka Segmentation

It is a challenging issue to accurately analyze the morphology of products in wire electrical explosion. Distinguished from conventional electron microscope images, the images of the electric explosion products show different contrast and color. Since the spraying process may contain various phases of copper (liquid, gas, and plasma) at the same time, the morphology of the final solidified product presents a non-uniform feature. The surface of some products is rough and black, and the surface of other products is smooth and metallic. Using a fixed threshold to binarize the image directly will cause a huge error in the analysis. The Trainable Weka Segmentation is a Fiji plugin that combines a collection of machine learning algorithms with a set of selected image features to produce pixel-based segmentations. It uses a manual drawing tool to calibrate areas of interest in the image. The software learns and trains from the input calibration area. It continues to correct and iteratively train again. Finally, the accurate calibration of the shape of different contrast electrical explosion products was achieved.

3. Results

3.1. Pulse Current Curve Parametric Statistics

The main parameters’ peak impulse current, I_p, and half-discharge period, T/2, are shown in

Table 5. Peak impulse current and half-discharge period with different diameters and hollow ratios.

Diameter D/mm	Hollow Ratio a (d/D)	Peak Impulse Current Ip/kA	Half-Discharge Period T/2/μs
0.3	0	14.59	2.00
	0.5	13.02	1.95
	0.7	10.89	1.92
0.4	0	18.63	2.26
	0.5	17.34	2.23
	0.7	14.73	2.22
0.5	0	22.64	2.50
	0.5	21.42	2.42
	0.7	17.91	2.40

. I_p and T/2 are physical quantities determined by measuring the instrument directly. The Rogowski coil, ranging from 0 to 50 kA, usually reads to two decimal places. The peak impulse current is used to measure the intensity of energy deposition. When the load wire is in three diameters and three hollow degrees of loading conditions, the peak impulse current value of electric explosion ranges from 10.89 to 22.64 kA. The load wire deposits energy and explodes electrically during the half-discharge period, T/2, which ranges from 1.92 to 2.54 μs. Therefore, the characteristic frequency range of the effective deposition period of the wire electrical explosion is about 200 to 260 kHz. The variation trend of the peak impulse current and half-discharge period is shown in Figure 3.

As shown in Figure 3, The peak impulse current, I_p, and the half-discharge period, T/2, both increase with the diameter and decrease with the hollow ratio. When the hollow ratio, a, is constant, the explosive inflection point of the discharge current curve appears later with the increase of the wire diameter. When the outer diameter, D, is constant, the explosive inflection point of the discharge current curve appears earlier with the increase of the wire hollow ratio [30]. The optimal wire should be thick and hollow.

3.2. Morphological Distribution of the Product

When Cu wire with a diameter of 0.5 mm is electrically exploded, the electrical explosive products of the Cu wire with different hollow ratios are collected at a calibrated distance of 12 mm. The morphology of the electrical explosive products is shown in Figure 4.

There are product particles with tiny particle sizes regardless of the hollow ratios of the Cu wire. However, the product’s average particle size decreases with the increase of the hollow ratio. Meanwhile, the particle size has a better homogeneous degree when the hollow ratio.is 0.5. The Weka neural network algorithm was used to identify the particle boundary of the spray product image. The calibration and statistics of spraying particles are shown in Figure 5.

As shown in Figure 5a, the complexity of the boundary contour of the spraying product varies with the hollow ratio of the wire. The shape of the solid wire explosion product is a radial polygon. With the increase of the hollow ratio of the wire, the boundary contour of the product tends to be a regular convex polygon. According to Figure 5b, the number of product particles in the concerned area of 0~1000 μm² is counted. When the hollow degree is 0.7, the size of the product particle is more concentrated than that when the hollow degree is 0.5. According to Figure 5c, the number of product particles in the entire area is counted. With the increase of the hollow ratio of the wire, the maximum area of the product particle decreases greatly.

The phase state and energy utilization degree of wire electrical explosive products are deduced by the particle morphology and quantity distribution. When the driving energy makes the wire ionized ideally, the product should be uniform-sized nanoparticles. When the driving energy does not ionize the wire uniformly, there will be plasma, vapor, and liquid droplets in the product. The product will no longer be uniform, and large agglomeration particles will appear. When the inhomogeneity of the ionization of the wire is intensified, the product will appear a significant droplet spray shape. The more complex the phase components in the product is, the lower is the energy utilization that can be qualitatively considered.

3.3. Peak Overpressure of the Shockwave

The pressure sensor determines the amplitude of the incident shockwave. When driven by three times the theoretical enthalpy value, the peak overpressure of the shockwave with three hollow ratios and three diameters of Cu wires are shown in Figure 6a,b. The typical pressure waveform is shown in Figure 6c.

As shown in Figure 6a, the diameter and hollow ratio of the wire have different intensity and form effects on the peak overpressure of the shockwave. In the study on the hollow ratio of Cu wire, the pressure soars with the increase of the hollow ratio when the diameter is a constant value. In the study on the diameter of Cu wire, the pressure decreases with the increase of the wire diameter when the hollow ratio, a, is 0. The pressure increases slightly with the increase of the wire diameter when the hollow ratio, a, is 0.5. The pressure increases significantly with the increase of the wire diameter when the hollow ratio, a, is 0.7. The peak overpressure of the shockwave ranges from 25.2 MPa to 46.1 MPa. A wire with a larger diameter and bigger hollow ratio produces a stronger shockwave when driven by triple enthalpy. As shown in Figure 6b, blue and gray bubbles represent peak overpressure and driving energy, respectively. The driving energy is triple the enthalpy of the wire with the diameter and hollow ratio. The relative size relationship between blue and gray bubbles represents the absolute conversion efficiency of driving energy to peak overpressure. The Figure 6c indicates that the third peak in a typical waveform represents the peak overpressure of the shockwave, while the first two peaks are electromagnetic interference.

The peak overpressure of the shockwave depends on the deposition efficiency of energy during the electrical explosion process of the Cu wire. The inhomogeneity of the heating phase transition limits the energy deposition efficiency. From the perspective of discharge energy, the uneven heating process prevents the copper wire from completely converting into plasma to deposit energy. The core position of the Cu wire can only be converted into metal droplets to consume the electric energy, which weakens the translation to the shockwave mechanical energy. From the perspective of discharge time, the ionization of the wire would occur on its surface firstly, and gradually evolve to the wire’s core layer by layer. The non-uniform phase transition of the wire in space leads to the prolongation of the effective discharge time. Ultimately, the energy conversion power is reduced.

4. Discussion

4.1. Research on Solid Wire and the Extension of Hollow Wire

According to the integral state of Maxwell’s equations, the current density distribution, J(r), and the corresponding skin depth, δ, on the wire section are expressed by the Bessel function [23,31]. When studying the δ and the J(r) of wire in the electric explosion process, a new expression related to wire structure is needed. Skin depth and current density can be expressed by new synthetic formulas respectively, as shown in Equations (5) and (6). $r\in \left[-D/2,D/2\right]$ is the radius range, f is the characteristic frequency of an electric explosion discharge loop, σ is the conductivity of the Cu wire, μ_o is the vacuum permeability, μ_r is the relative permeability, J₀ is the theoretical total current phasor, D is the outer diameter of a wire, j is the complex unit, Ψ₀ is the zero order of the first kind of Bessel function, and Ψ₁ is the first order of the first kind of Bessel function.

$$\delta =\sqrt{\frac{1}{\pi f\sigma {\mu }_0{\mu }_r}}$$

(5)

$$J\left(r\right)=J_0\frac{\pi D\left(1-j\right)}{\delta }\frac{{\Psi }_0\left[\left(1-j\right)r/\delta \right]}{{\Psi }_1\left[\left(1-j\right)D/2\delta \right]}$$

(6)

According to Table 5 and Equations (5) and (6), when the energy pulse is discharged in a copper wire with a diameter of 0.5 mm, the normalized calculation results of the current density distribution in the horizontal section are shown in Figure 7a. The value of the shadow is the normalized current value, which decays inward on the radial normal. The current density at the core of the wire decays to about 0.2 times the size of its surface. The normalized current density distribution and skin depth in the vertical section of wires with different diameters are shown in Figure 7b. Here, m is the ratio of the current density at the radius, r, to that at the wire’s surface, m = J(r)/J(D/2), and δ₁, δ₂, and δ₃ are the skin depths of solid wire with three diameters. The results show that the current density at the wire core decays to 0.20~0.35 times the size of its surface, with δ ranging from 0.136 mm to 0.147 mm. It is indicated that the non-uniform attenuation effect of the current is more pronounced in larger-diameter wires. The attenuation leads to uneven development of the electric explosion of wire.

When the core of the wire is hollowed out, the current density still satisfies the integral form of Maxwell’s equations. It means that the current density is still distributed as a Bessel function in the non-hollow part of the wire. The proportional relationship of current densities at different r does not change. What changes is the absolute value of the current density. The deductions of the normalized current density in the hollow wire are shown in Figure 8a,b. Interceptive current density is 0.45~0.53 times the size of its surface when the hollow ratio is 0.5. Moreover, it is 0.66~0.70 times larger when the hollow ratio is 0.7.

The J_a(r) is the simplified expression of the current density when the hollow ratio of wire is a. It is defined as J₀(r), J_0.5(r), and J_0.7(r) when a is 0, 0.5, and 0.7, respectively. The relation between J_a(r) satisfies Equation (7).

$$I={{\displaystyle \int }}_0^{0.5D}{J}_0\left(r\right)2\pi rdr={{\displaystyle \int }}_{0.25D}^{0.5D}{J}_{0.5}\left(r\right)2\pi rdr={{\displaystyle \int }}_{0.35D}^{0.5D}{J}_{0.7}\left(r\right)2\pi rdr$$

(7)

Due to the integral interval variation, the hollow wire’s current density is amplified in proportion to coefficient, K_a, when the hollow ratio of wire is a. The current density at this point is rewritten as $J_a\left(r\right)=K_a{\Psi }_0\left(r/\delta \right)$. The integral of current density in Equation (7) satisfies the recursive relation as $d\left[r{\Psi }_1\left(r\right)\right]/dr=r{\Psi }_0\left(r\right)$. The Cu wire current coefficients, K_a, of the three hollow ratios in the integral identity are K₀, K_0.5, and K_0.7. The proportional relationship among the three is Equation (8).

$$\begin{gathered} 0.958K_0=0.642K_{0.5}=0.392K_{0.7} \\ {K}_0:K_{0.5}:K_{0.7}\approx 2:3:5 \end{gathered}$$

(8)

According to Equation (8), K_a represents the amplification factor of Cu wire current density with different hollow ratios. According to the above conclusion, when the hollow ratio of the wire is 0.5, the current density flowing through the remaining Cu wire is amplified to about 1.5 times the solid wire current density. When the hollow ratio of the wire reaches 0.7, the current density is amplified to about 2.5 times that of the solid wire. In the wire electric explosion test under the condition of hollower parameters, the current amplitude increases significantly. It is conducive to the more rapid explosion of the wire and makes the phase distribution in the process of explosion of the wire more compact. Increasing the wire diameter and hollow ratio is more conducive to the deposition energy release to the peak overpressure of a shockwave.

4.2. Inference of Phase States Distribution and Peak Overpressure of Shockwave

Figure 3, Figure 5, and Figure 6, respectively, show the test and analysis results for the pulse current, the explosion product, and the peak overpressure. The experimental results were compared with the calculated results of the skin effect in Equation (8). The extreme simplification and qualitative derivation of the phase distribution of a wire electric explosion is presented in Figure 9.

As shown in Figure 9a, the current density, J(r), showed a Bessel function distribution. The wire is heated unevenly in the radial direction because of the impulse current distribution. Therefore, the wire will rapidly undergo a phase state transition. Under the spatial structure properties (diameter and hollow ratio), the temperature inhomogeneity caused by phase state components’ coexistence is dynamically changing. They may exist simultaneously in the solid–liquid–gas–plasma state, or in two of the four states. The study of the characteristic parameters of temperature and resistance values is always a difficult issue in the field of electrical explosion. The signals of temperature and resistance change too fast and the amplitude is very large. They are far beyond the range of existing sensors. The characteristic parameters can only be inferred qualitatively from existing theoretical relations. The following change processes and mechanisms can be derived from the theoretical relationships.

The sequence of phase transitions in the radial direction of the wire is shown in Figure 9a. The plasma’s heat is dissipated radially outwards and transferred inward to heat the inner wire. The heavy cations pinch inward to concentrate energy. The high temperature and pressure of the Cu plasma and water vapor are compressed outward into a shockwave. As shown in Figure 9b, it can be defined as the parallel model of Cu wires in different phase states because of its spatial inhomogeneity during the electrical explosion. When the current density of the Bezier distribution drives the phase state components of a specific mass fraction, the resistance and temperature values are different. As shown in Figure 9c, the resistance value of the Cu wire is closely related to temperature. The resistance value of Cu increases first and then decreases with the increase of its temperature. The resistance value increases significantly at the point of the phase transition temperature of the Cu wire and decreases to the minimum value instantaneously when ionization occurs. The positive feedback mechanism results in a more intense Bessel function distribution in the radial direction. The collected products of small particle size may be the high-temperature plasma during the electrical explosion. Moreover, the coarse particle size may be the splashed metal droplets. The fine and the coarse particles may come from the wire’s surface and core, respectively. The heat conduction accompanying the phase distribution has two effects on the subsequent evolution of the wire. For the surface of the wire, the dissipative heat conduction wastes the deposited energy in the plasma. For the core of the wire, the dual action of heat conduction and internal current density can vaporize the core Cu wire in the subsequent time.

In the wire electric explosion test, the essence of increasing shockwave energy is to improve energy conversion efficiency. When the wire is heated by an impulse current, the skin effect causes the transition of phase states inside the wire to be uneven in time and space. Finally, the energy conversion efficiency of electrical energy to mechanical energy of the shockwave is reduced. Therefore, the hollow wires of different structures solve this effect. It can be found that increasing the hollow ratio of the wire can greatly alleviate this spatial–temporal inhomogeneity. Because the hollow wire has a higher energy specific surface area, the phase states distribution of the wire is more concentrated in the process of energy conversion. The energy of the shockwave is finally improved.

5. Conclusions

The peak overpressure of the shockwave is related to the impulse current temporal structure and the Cu wire spatial structure. On the spatial scale, the different phase state Cu wires are coaxial hollow cylinders nested with each other between the electrodes. In the time scale, the ionizing ablation of the Cu wire evolves from the surface to the inside layer by layer along the radius direction.

The morphology and quantity of products indicate that it is going to be uniform gradually with the hollow ratio rising from 0 to 0.7. It is a parallel model of multi-phase Cu wires nested into each other proved by the distribution during the wire electrical explosion.

The range of peak overpressure is 25.2~47.7 MPa. The peak overpressure soars with the increase of the hollow ratios when the diameter is a constant. The correlation between shockwave and wire diameter changes from negative to positive with the increase of the hollow ratio from 0 to 0.7. The Cu wire with a hollow ratio of 0.5 and a diameter of 0.5 mm is a stable and efficient electrical explosive material.

When the circuit parameters are controllable and fixed, the hollow cylindrical copper wire is beneficial to alleviate the influence of skin effect and to obtain ideal peak overpressure of the shockwave.